Properties

Label 41T6
Order \(410\)
n \(41\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{41}:C_{10}$

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Group action invariants

Degree $n$ :  $41$
Transitive number $t$ :  $6$
Group :  $C_{41}:C_{10}$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41), (1,25,10,4,18,40,16,31,37,23)(2,9,20,8,36,39,32,21,33,5)(3,34,30,12,13,38,7,11,29,28)(6,27,19,24,26,35,14,22,17,15)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 10, 10, 10, 10, 1 $ $41$ $10$ $( 2, 5,17,24,11,41,38,26,19,32)( 3, 9,33, 6,21,40,34,10,37,22)( 4,13, 8,29,31, 39,30,35,14,12)( 7,25,15,16,20,36,18,28,27,23)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,11,19,17,38)( 3,21,37,33,34)( 4,31,14, 8,30)( 5,41,32,24,26) ( 6,10, 9,40,22)( 7,20,27,15,18)(12,29,35,13,39)(16,28,25,36,23)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,17,11,38,19)( 3,33,21,34,37)( 4, 8,31,30,14)( 5,24,41,26,32) ( 6,40,10,22, 9)( 7,15,20,18,27)(12,13,29,39,35)(16,36,28,23,25)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,19,38,11,17)( 3,37,34,21,33)( 4,14,30,31, 8)( 5,32,26,41,24) ( 6, 9,22,10,40)( 7,27,18,20,15)(12,35,39,29,13)(16,25,23,28,36)$
$ 10, 10, 10, 10, 1 $ $41$ $10$ $( 2,24,38,32,17,41,19, 5,11,26)( 3, 6,34,22,33,40,37, 9,21,10)( 4,29,30,12, 8, 39,14,13,31,35)( 7,16,18,23,15,36,27,25,20,28)$
$ 10, 10, 10, 10, 1 $ $41$ $10$ $( 2,26,11, 5,19,41,17,32,38,24)( 3,10,21, 9,37,40,33,22,34, 6)( 4,35,31,13,14, 39, 8,12,30,29)( 7,28,20,25,27,36,15,23,18,16)$
$ 10, 10, 10, 10, 1 $ $41$ $10$ $( 2,32,19,26,38,41,11,24,17, 5)( 3,22,37,10,34,40,21, 6,33, 9)( 4,12,14,35,30, 39,31,29, 8,13)( 7,23,27,28,18,36,20,16,15,25)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1 $ $41$ $5$ $( 2,38,17,19,11)( 3,34,33,37,21)( 4,30, 8,14,31)( 5,26,24,32,41) ( 6,22,40, 9,10)( 7,18,15,27,20)(12,39,13,35,29)(16,23,36,25,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $41$ $2$ $( 2,41)( 3,40)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,32)(12,31) (13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)$
$ 41 $ $10$ $41$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41)$
$ 41 $ $10$ $41$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41, 2, 4, 6, 8, 10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40)$
$ 41 $ $10$ $41$ $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37,40, 2, 5, 8,11,14,17,20,23,26,29,32, 35,38,41, 3, 6, 9,12,15,18,21,24,27,30,33,36,39)$
$ 41 $ $10$ $41$ $( 1, 7,13,19,25,31,37, 2, 8,14,20,26,32,38, 3, 9,15,21,27,33,39, 4,10,16,22, 28,34,40, 5,11,17,23,29,35,41, 6,12,18,24,30,36)$

Group invariants

Order:  $410=2 \cdot 5 \cdot 41$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [410, 1]
Character table:   
      2  1   1   1   1   1   1   1   1   1  1   .   .   .   .
      5  1   1   1   1   1   1   1   1   1  1   .   .   .   .
     41  1   .   .   .   .   .   .   .   .  .   1   1   1   1

        1a 10a  5a  5b  5c 10b 10c 10d  5d 2a 41a 41b 41c 41d
     2P 1a  5b  5c  5a  5d  5d  5a  5c  5b 1a 41b 41a 41d 41c
     3P 1a 10b  5b  5d  5a 10d 10a 10c  5c 2a 41c 41d 41b 41a
     5P 1a  2a  1a  1a  1a  2a  2a  2a  1a 2a 41b 41a 41d 41c
     7P 1a 10c  5c  5a  5d 10a 10d 10b  5b 2a 41c 41d 41b 41a
    11P 1a 10a  5a  5b  5c 10b 10c 10d  5d 2a 41c 41d 41b 41a
    13P 1a 10b  5b  5d  5a 10d 10a 10c  5c 2a 41c 41d 41b 41a
    17P 1a 10c  5c  5a  5d 10a 10d 10b  5b 2a 41d 41c 41a 41b
    19P 1a 10d  5d  5c  5b 10c 10b 10a  5a 2a 41d 41c 41a 41b
    23P 1a 10b  5b  5d  5a 10d 10a 10c  5c 2a 41a 41b 41c 41d
    29P 1a 10d  5d  5c  5b 10c 10b 10a  5a 2a 41c 41d 41b 41a
    31P 1a 10a  5a  5b  5c 10b 10c 10d  5d 2a 41a 41b 41c 41d
    37P 1a 10c  5c  5a  5d 10a 10d 10b  5b 2a 41a 41b 41c 41d
    41P 1a 10a  5a  5b  5c 10b 10c 10d  5d 2a  1a  1a  1a  1a

X.1      1   1   1   1   1   1   1   1   1  1   1   1   1   1
X.2      1  -1   1   1   1  -1  -1  -1   1 -1   1   1   1   1
X.3      1   A -/A  -B -/B  /B   B  /A  -A -1   1   1   1   1
X.4      1   B -/B -/A  -A   A  /A  /B  -B -1   1   1   1   1
X.5      1  /B  -B  -A -/A  /A   A   B -/B -1   1   1   1   1
X.6      1  /A  -A -/B  -B   B  /B   A -/A -1   1   1   1   1
X.7      1 -/A  -A -/B  -B  -B -/B  -A -/A  1   1   1   1   1
X.8      1 -/B  -B  -A -/A -/A  -A  -B -/B  1   1   1   1   1
X.9      1  -B -/B -/A  -A  -A -/A -/B  -B  1   1   1   1   1
X.10     1  -A -/A  -B -/B -/B  -B -/A  -A  1   1   1   1   1
X.11    10   .   .   .   .   .   .   .   .  .   C   D   F   E
X.12    10   .   .   .   .   .   .   .   .  .   D   C   E   F
X.13    10   .   .   .   .   .   .   .   .  .   E   F   C   D
X.14    10   .   .   .   .   .   .   .   .  .   F   E   D   C

A = -E(5)
B = -E(5)^2
C = E(41)^3+E(41)^7+E(41)^11+E(41)^12+E(41)^13+E(41)^28+E(41)^29+E(41)^30+E(41)^34+E(41)^38
D = E(41)^6+E(41)^14+E(41)^15+E(41)^17+E(41)^19+E(41)^22+E(41)^24+E(41)^26+E(41)^27+E(41)^35
E = E(41)+E(41)^4+E(41)^10+E(41)^16+E(41)^18+E(41)^23+E(41)^25+E(41)^31+E(41)^37+E(41)^40
F = E(41)^2+E(41)^5+E(41)^8+E(41)^9+E(41)^20+E(41)^21+E(41)^32+E(41)^33+E(41)^36+E(41)^39